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Mirrors > Home > ILE Home > Th. List > nltmnf | GIF version |
Description: No extended real is less than minus infinity. (Contributed by NM, 15-Oct-2005.) |
Ref | Expression |
---|---|
nltmnf | ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 7899 | . . . . . . 7 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2421 | . . . . . 6 ⊢ ¬ -∞ ∈ ℝ |
3 | 2 | intnan 915 | . . . . 5 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) |
4 | 3 | intnanr 916 | . . . 4 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) |
5 | pnfnemnf 7911 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
6 | 5 | nesymi 2370 | . . . . 5 ⊢ ¬ -∞ = +∞ |
7 | 6 | intnan 915 | . . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ = +∞) |
8 | 4, 7 | pm3.2ni 803 | . . 3 ⊢ ¬ (((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) |
9 | 6 | intnan 915 | . . . 4 ⊢ ¬ (𝐴 ∈ ℝ ∧ -∞ = +∞) |
10 | 2 | intnan 915 | . . . 4 ⊢ ¬ (𝐴 = -∞ ∧ -∞ ∈ ℝ) |
11 | 9, 10 | pm3.2ni 803 | . . 3 ⊢ ¬ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ)) |
12 | 8, 11 | pm3.2ni 803 | . 2 ⊢ ¬ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))) |
13 | mnfxr 7913 | . . 3 ⊢ -∞ ∈ ℝ* | |
14 | ltxr 9660 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))))) | |
15 | 13, 14 | mpan2 422 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 < -∞ ↔ ((((𝐴 ∈ ℝ ∧ -∞ ∈ ℝ) ∧ 𝐴 <ℝ -∞) ∨ (𝐴 = -∞ ∧ -∞ = +∞)) ∨ ((𝐴 ∈ ℝ ∧ -∞ = +∞) ∨ (𝐴 = -∞ ∧ -∞ ∈ ℝ))))) |
16 | 12, 15 | mtbiri 665 | 1 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 2125 class class class wbr 3961 ℝcr 7710 <ℝ cltrr 7715 +∞cpnf 7888 -∞cmnf 7889 ℝ*cxr 7890 < clt 7891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-cnex 7802 ax-resscn 7803 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-rab 2441 df-v 2711 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-xp 4585 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 |
This theorem is referenced by: mnfle 9677 xrltnsym 9678 xrlttr 9680 xrltso 9681 xltnegi 9717 xposdif 9764 qbtwnxr 10135 xrmaxiflemab 11121 xrmaxltsup 11132 xrbdtri 11150 blssioo 12884 |
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